Question

With usual notation a function has minimum at (a, b), if :​

Answer 1

With usual notation, a function $f(x)$ minimum in $(a,b)$ only if there exists a  point $c$ in $(a,b)$ such that $f'(c)=0$ and $f''(c) > 0$.

• The derivative of a function at a point gives us the rate of change of the function at that point.
• At any point of minimum or maximum, the function which is decreasing reaches the lowest point and then increases or the function which is increasing reaches the highest point and then decreases.
• During increasing and decreasing of the function, the derivative is positive and negative respectively.
• At the highest or lowest point, i.e. maximum or minimum the derivative is zero.
• Since at the minimum point, the derivative changes from negative to positive, the second derivative i.e the rate of change of derivative is positive.